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122 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 6, NO. 1, FEBRUARY 2017
Performance Analysis of OFDM-Based Nonlinear AF Multiple-Relay Systems
Nagendra Kumar, Sanjeev Sharma, and Vimal Bhatia, Senior Member, IEEE
Abstract—We derive closed-form expressions of outage
probability with asymptotic analysis for orthogonal frequency
division multiplexing-based nonlinear amplify-and-forward
multiple-relay systems using best-relay selection over indepen-
dent and non-identically distributed Nakagami-mfading channels
for integer values of fading parameter. We then investigate a
novel closed-form expression of average symbol error rate for
a general order rectangular quadrature amplitude modulation
scheme by using the well-known cumulative distribution function-
based approach. Numerically evaluated results are compared
with Monte Carlo simulations and the accuracy of the derived
expressions is verified.
Index Terms—Multiple-relay, outage probability, ASER, non-
linear amplifier, RQAM, Nakagami-m.
I. INTRODUCTION
COOPERATIVE relaying has received significant research
interest in current and future wireless communication
systems in view of their ability to enhance coverage, reliabil-
ity and spectral efficiency. Cooperative relaying network with
orthogonal frequency division multiplexing (OFDM) technol-
ogy has been extensively studied in the literature (see [1]–[4]
and references therein). The basic concept behind the use of
OFDM technology in wireless relaying network is to pro-
vide high coverage and large throughput in LTE-Advanced,
future 5G and beyond communication networks. Amplify-and-
forward (AF) relay networks are particularly more attractive
due to their low complexity, and are thus used for the sce-
nario where systems have limited signal processing resources.
Nakagami-mis well-known versatile statistical distribution,
used to model a variety of fading environments [3], [5], [6].
It is well-known that rectangular quadrature amplitude mod-
ulation (RQAM) is generic modulation scheme which includes
square QAM (SQAM), binary phase-shift keying (BPSK),
orthogonal binary frequency-shift keying (OBFSK), quadra-
ture phase-shift keying (QPSK) and multilevel amplitude-shift
keying (ASK) modulation schemes as special cases, and due
to its high bandwidth efficiency characteristics, it is preferred
in LTE-Advanced standards [5], [6].
With move towards 5G communication, a large bandwidth is
required for multimedia transmission over wireless channels,
and designing a linear amplifier for large bandwidth system
is highly difficult. In cooperative relay networks, the received
Manuscript received November 15, 2016; accepted December 11,
2016. Date of publication December 14, 2016; date of current version
February 16, 2017. The associate editor coordinating the review of this paper
and approving it for publication was A. Kammoun.
The authors are with the Discipline of Electrical Engineering,
Indian Institute of Technology Indore, Indore 453441, India (e-mail:
phd1301202008@iiti.ac.in).
Digital Object Identifier 10.1109/LWC.2016.2639498
OFDM signal with high peak-to-average power ratio (PAPR)
at the relay is much more susceptible to distortions caused
by nonlinear amplifier. Recently, researchers are interested in
studying effects of nonlinear amplifier on the performance of
OFDM-based AF relay networks. In [1], outage probability
(OP) and average symbol error rate (ASER) expressions have
been derived over Rayleigh fading channels for nonlinear AF
relay network considering maximal ratio combining (MRC)
and selection combining (SC) scheme for single relay and
a direct link. In [2], closed-form OP expressions have been
derived in two-way nonlinear fixed and variable gain single
AF relay network over Rayleigh fading channels. Considering
Nakagami-mdistributed environments, closed-form expression
of OP using MRC in [3], and closed-form expressions of OP
and ASER using SC in [4] have been derived for the similar
system model as in [1]. On the other hand, researchers are
also interested in ASER performance for RQAM in various
relaying and non-relaying wireless communication networks
(see [5], [6] and the references therein).
In this letter, for the first time an OFDM-based nonlinear
AF multiple-relay network for independent and non-identically
distributed (i.n.i.d.) Nakagami-menvironments with modified
fading and spreading parameters, wherein best-relay selection
scheme is used for Knumbers of relay and MRC scheme is
considered for the best relay and a direct link. We evaluate
closed-form expressions of OP, asymptotic outage behavior
in the high signal-to-noise ratio (SNR) regime, and ASER
for general order RQAM, which are valid for integer value
of fading parameters. To the best of authors’ knowledge the
derived expressions and the important aspects covered in this
letter are not available in the literature. Because of using spec-
trally efficient relaying network with nonlinear amplifier and
higher order modulation scheme in the considered system over
a versatile channel, the work covered in this letter will be of
immense interest in 5G and beyond communication systems.
II. SYSTEM AND CHANNEL MODEL
We consider OFDM-based nonlinear AF multiple-relay
communication system which consists of one source node
S, one destination node D, and Krelay nodes Rk, where
k=1,2,...,K. The communication between Sand Doccurs
through a direct link S→Dand Kindirect links S→Rk→D
over i.n.i.d frequency selective Nakagami-mfaded channel. It
is assumed that all the nodes are equipped with single antenna,
all communication between the nodes operate in half duplex
mode, all the nodes are synchronized at symbol level and
each relay node consists of a nonlinear amplifier. To avoid
inter-symbol interference, the length of cyclic prefix (CP) is
considered greater than or equal to channel impulse response.
2162-2345 c
2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
KUMAR et al.: PERFORMANCE ANALYSIS OF OFDM-BASED NONLINEAR AF MULTIPLE-RELAY SYSTEMS 123
Transmitted power for the source and the relay’s subcarri-
ers are Psand Pr, respectively, and assumed to be same for
each subcarriers. Full channel state information is assumed to
be known at the relays. By using best-relay selection algo-
rithm [7], the upper-bound SNR for nth subcarrier at Dcan be
given as
n
t≤n
SD +argmaxk∈{1,...,K}n
k,1≤n≤N,(1)
where n
SD =Ps|hn
SD|2/σ 2
vis SNR for S→Dlink, n
k=
min(n
SRk,
n
RkD,
NA)[1], wherein, n
SRk=Ps|hn
SRk|2/σ 2
v
and n
RkD=Pr|hn
RkD|2|KR
0|2/σ 2
vare SNRs for S→Rkand
Rk→Dlinks, respectively. Also, NA =Pr|KR
0|2/σ 2
dRis the
instantaneous SNR at the output of nonlinear amplifier and
Nrepresents number of subcarriers. Further, hn
SD,hn
SRkand
hn
RkDare the channel frequency response of the corresponding
links, and σ2
vis the variance of additive white Gaussian noise
(AWGN). The closed-form expressions of KR
0and σ2
dRare
given in [3] which represent fixed gain and nonlinear distor-
tion variance for amplifier, respectively. =Ps/σ 2
vrepresents
average transmit SNR.
III. OUTAGE PROBABILITY ANALYSIS
The OP of the nth subcarrier for a given threshold γth can
be formulated as
Pn
out(γth )=Pr [n
SD +n
k∗<γ
th],
=∞
0fn
SD (x)Fn
k∗(γth −x)dx,(2)
where n
k∗=argmaxk∈{1,...,K}n
kis SNR of the best relay
(k∗), fX(·)and FX(·)represent probability density function
(PDF) and CDF of the corresponding random variable (RV).
Further, Fn
k∗(γth)can be given as
Fn
k∗(γth)=
K
k=11−1−Fn
SRk(γth)1−FNA (γth)
×1−Fn
RkD(γth).(3)
Generally, Nakagami-mdistribution is considered for RV
in time domain with fading and spreading parameters ml
and lfor lth tap, respectively. However, for frequency
domain RV, time domain Nakagami-mis approximated by
another Nakagami- ˜mdistribution with modified fading and
spreading parameters ˜mand ˜
, respectively, where ˜m=
(L−1
l=0l)2/(L−1
l=0
2
l
ml+L−1
l=0L−1
l1=0,l=l1ll1)and ˜
=
L−1
l=0l, where Lis total number of independent chan-
nel taps [8], [9]. Further, the PDF and CDF expressions
for Nakagami-mdistribution with modified parameters can
be expressed as fX(x)=(˜m
˜
)˜mx˜m−1
[˜m]e−˜mx
˜
u(x)and FX(x)=
1
[˜m]ϒ( ˜m,˜mx
˜
)u(x), where ϒ(·)and (·)represent the lower
incomplete and the complete Gamma functions [10], and u(·)
is unit step function. Moreover, SNR for nonlinear amplifier
is assumed to be fixed as it is considered that the variations
of amplifier’s parameters are much slower than the wireless
channel variations. Thus, FNA (γth)=u(γth −¯
NA), where
¯
NA =E[NA], wherein E[·] represents the statistical average
Fig. 1. ASER performance versus transmit SNR with linear and nonlin-
ear amplifiers at different RQAM constellations for mSD =1,mSRk=1,
mRkD=1, K=1andβ=1.
operator. To evaluate Fn
k∗(γth)in (3), substitute FNA (γth)
and CDF of the corresponding links into (3), then applying
[10, (8.352.2)], and binomial and multinomial expansions as
in [7]. Further, invoking expressions of Fn
k∗(γth −x)and
fn
SD (x)into (2) and performing the required integration, we
get
Pn
out(γth )=1
( ˜mSD)ϒ˜mSD ,˜mSDγth
¯
SD +R0γσi
th e−γth
¯
e
ϒ˜mSD +t,γth
¯
E−ϒ˜mSD +t,γmax
¯
E,(4)
where, R0=1
( ˜mSD)(˜mSD
¯
SD )˜mSD K
k=1K
kk(˜mSRk−1)
i=0k(˜mRkD−1)
j=0
αk
iαk
ji+j
t=0i+j
t(−1)k+t(˜mSRk
¯
SRk
)i(˜mRkD
¯
RkD)j(1
¯
E)−(˜mSD+t), and
σi=i+j−twherein, ˜mSD,˜mSRkand ˜mRkDare mod-
ified fading parameters of the corresponding links. The
modified spreading parameters for the corresponding links
are represented as ˜
SD =E[|hn
SD|2], ˜
SRk=E[|hn
RkD|2]
and ˜
RkD=E[|hn
RkD|2], respectively. Also, αν
μis the
coefficient in corresponding expansion which can be cal-
culated recursively using [10, (0.314)] as αν
0=(θ0)ν,
αν
1=ν(θ1)and αν
ν( ˜m0−1)=(θ ˜m0−1)νfor 0 ≤μ≤ν( ˜m0−1),
αν
μ=1
μθ0μ
q=1[qν−μ+q]θqαν
μ−qfor 2 ≤μ≤˜m0−1, and
αν
μ=1
μθ0˜m0−1
q=1[qν−μ+q]θqαν
μ−qfor ˜m0≤μ≤ν( ˜m0−1),
with θμ=1
μ!and ˜m0∈{˜mSRk,˜mRkD}. Further, σi=i+j−t,
¯
E=(˜mSD
¯
SD −k˜mSRk
¯
SRk−k˜mRkD
¯
RkD)−1,¯
e=(k˜mSRk
¯
SRk+k˜mRkD
¯
RkD)−1
and γmax =max(γth −¯
NA,0), wherein, ¯
SD,¯
SR and ¯
RD
represent the statistical expectation of n
SD,n
SR and n
RD,
respectively.
IV. ASYMPTOTIC OUTAGE ANALYSIS
We now obtain the OP expression in the high SNR regime
(¯
→∞)to examine a useful insight into system’s diversity
order. Substituting FNA(γth )and CDF of the corresponding
links into (3), then making use of approximation ϒ(υ,z)≈
z→0
zυ
υ, and discarding higher order terms. Invoking fn
SD (x)and
simplified expressions of Fn
k∗(γth −x)for high SNR into (2),
and then performing the required integration followed by high
124 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 6, NO. 1, FEBRUARY 2017
Fig. 2. Outage (left) and ASER (right) performances versus transmit
SNR with variation of number of relays at different thresholds and different
constellations, respectively.
SNR approximation, we can get asymptotic OP expression for
Case (A):- γth >¯
NA as
Pn
out(γth )≈
¯
→∞
˜mSDγth
¯
SD ˜mSD
( ˜mSD +1)1−¯
NA
γth ˜mSD .(5)
Similarly, the asymptotic OP expression for Case (B):- γth <
¯
NA,isgivenin(6), shown at the top of the next page.
Remarks: By comparing (5) and (6), we infer that the system
can achieve a diversity order of Gd=˜mSD for Case (A),
which depends only on fading parameter of the direct link and
independent from the number of relays, however, for Case (B)
the system can achieve a better diversity order of Gd=˜mSD +
Kmin(˜mSRk,˜mRkD), which depends on fading parameter of all
the links as well as the number of relays.
V. ASER P ERFORMANCE ANALYSIS
By using well-known CDF based approach, an expression of
ASER for several modulation schemes can be obtained as [5]
PASER(e)=−∞
0
P
s(e|)F()d, (7)
where P
s(e|) denotes the first derivative of the conditional
SER with respect to instantaneous SNR () for a modula-
tion scheme in AWGN channels, and F() is the CDF for
the received SNR which can be obtained by replacing γth by
in (4). The first derivative of the conditional SER expres-
sion for M-ary RQAM scheme in AWGN channel is given
in [5] and [6]as
Ps(e|) =ap(q−1)
√2π−1
2e−a2
2+b(p−1)q
√2π
×−1
2e−b2
2−abpq
πe−(a2+b2
2) 1F11;3
2;a2
2
−abpq
πe−(a2+b2
2) 1F11;3
2;b2
2,(8)
where M=MI×MQ, wherein, MIand MQare the number
of in-phase and quadrature-phase constellation points, respec-
tively, p=1−1
MI,q=1−1
MQ,a=6
(M2
I−1)+(M2
Q−1)β2,
b=βaand β=dQ/dI, in which, dQand dIdenote
quadrature and in-phase decision distance, respectively.
Invoking (4) and (8)into(7), applying some mathemat-
ical simplifications, and then performing the required
integration with the aid of [10, (6.455.2)], closed-form
expression of ASER for RQAM scheme is given as (9)
in the top of the next page, where, (μ,ν,δ,x,y)=
(μ) ˜mSD+ν( ˜mSD +1/2+δ)
(˜mSD+ν )((x2+y2)/2+˜mSD/¯
SD)˜mSD +1/2+δ2F1(1,˜mSD +1/2+
δ;˜mSD +1+ν;μ/((x2+y2)/2+˜mSD/¯
SD)),
(τ, ϕ ) =(1)τ(ϕ)τ
(1.5)ττ!, wherein, 2F1(·,·;·; ·)and
()ω=( +ω)/ () represent Gauss hypergeo-
metric function and Pochhammer symbol, respectively [10].
The derived expression of ASER for RQAM, (9) is valid
for integer value of fading parameter ˜m. By considering
the special cases of RQAM, the expression of ASER for
SQAM can be derived by substituting MI=MQ=√M
and β=1in(9). Similarly, for MI=2, MQ=1, p=0.5,
q=0, a=√2 and β=0, we can derive the expression
for BPSK modulation scheme. For small constellation size
(i.e., large values of aand b), and small values of ¯
SD,¯
SRk
and ¯
RkD, the amplifier shows linear characteristic because
of the saturation characteristic of decaying exponential as
lim
z→+∞e−z=0(in(9)). Thus, by considering ¯
NA →∞in (9),
we obtain the ASER expression for linear amplifier.
VI. NUMERICAL AND SIMULATION RESULTS
In this section, we present Monte’ Carlo simulation (exact)
results to confirm the validity of derived closed-form expres-
sions. All the analytical results were calculated with good
accuracy by truncating the infinite series to 15 terms. Here, we
adopt an OFDM-based nonlinear AF multiple-relay systems
wherein total number of subcarriers N=64, CP length equal
to 16, and frequency selective Nakagami-mchannel with mod-
ified fading and spreading parameters considering total number
of independent channel taps L=16. Further, it can be veri-
fied that when L≥2 and 1/2≤ml<∞then 1/2≤˜m<2
[8], [9]. The analysis in this letter, therefore, focuses on a
bounded value of ˜mwhich covers a wide range of ml. Since,
the derived expressions are valid only for integer-valued fad-
ing parameters, thus, the results are analyzed for ˜m=1. A
nonlinear amplifier, modeled by soft clipping (soft limiter),
is considered at each relay with amplifier saturation ampli-
tude Asat =1. Furthermore, Ps=Pr=0.5, which leads
¯
NA =17.5 dB (expressions for KR
0and σ2
dR,usedtofindthe
value of ¯
NA,isgivenin[3] and [4]), and ˜
SD =2, ˜
SRk=1
and ˜
RkD=2 are considered.
Fig. 1shows comparison of ASER curves versus transmit
SNR for both linear and nonlinear amplifiers with various
constellation order considering ˜mSD =1,˜mSRk=1 and
˜mRkD=1 in the considered systems. We observe that the
theoretical and the simulated curves match well for all the
investigated cases which confirms accuracy of the derived
expression (9). Furthermore, as predicted in Section V,the
impact of nonlinearity on ASER performance is more signifi-
cant at medium and high SNRs. As we know that the impact
of nonlinearity on OP performance becomes significant when
γth >¯
NA, however, the amplifier shows linear characteristic
when γth <¯
NA, which have been already discussed in [3].
KUMAR et al.: PERFORMANCE ANALYSIS OF OFDM-BASED NONLINEAR AF MULTIPLE-RELAY SYSTEMS 125
Pn
out(γth )≈
¯
→∞
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˜mSDγth
¯
SD ˜mSD
( ˜mSD)˜mSRkγth
¯
SRkK˜mSRk
[( ˜mSRk+1)]KK˜mSRk
l=0K˜mSRk
l(−1)l
˜mSD+l,if ˜mSRk<˜mRkD,
˜mSDγth
¯
SD ˜mSD
( ˜mSD)˜mRkDγth
¯
RkDK˜mRkD
[( ˜mRkD+1)]KK˜mRkD
l=0K˜mRkD
l(−1)l
˜mSD+l,if ˜mSRk>˜mRkD,
1
( ˜mSD)˜mSDγth
¯
SD ˜mSD K
k=0K
k1
[( ˜mSRk+1)]K−k˜mSRkγth
¯
SRk(K−k)˜mSRk1
[( ˜mRkD+1)]k
˜mRkDγth
¯
RkDk˜mRkD(K−k)˜mSRk+k˜mRkD
l=0(K−k)˜mSRk+k˜mRkD
l(−1)l
˜mSD+l,if ˜mSRk=˜mRkD
(6)
Pn
ASER(e)=ap(1−q)
√2π1
( ˜mSD)( ˜mSD /¯
SD,0,0,a,0)+R0(1/¯
E,t,σ
i+t,a,0)−σi−1/2
λ1σi−1/2
λ1¯
σi−1/2−λ1
NA
e−(a2
2+1
¯
e)¯
NA (1/¯
E,t,t+λ1+1/2,a,0)+b(1−p)q
√2π1
( ˜mSD)( ˜mSD /¯
SD,0,0,0,b)+R0(1/¯
E,t,σ
i+t,0,b)
−σi−1/2
λ1σi−1/2
λ1¯
σi−1/2−λ1
NA e−(b2
2+1
¯
e)¯
NA (1/¯
E,t,t+λ1+1/2,0,b)+abpq
π1
( ˜mSD)∞
μ1=0
(μ1,a2/2)( ˜mSD /¯
SD,0,μ
1+1/2,a,b)+∞
μ2=0(μ2,b2/2)( ˜mSD /¯
SD,0,μ
2+1/2,a,b)+R0∞
μ3=0
(μ3,a2/2)(1/¯
E,t,σ
i+t+μ3+1/2,a,b)+∞
μ4=0(μ4,b2/2)(1/¯
E,t,σ
i+t+μ4+1/2,a,b)−
e−(a2+b2
2+1
¯
e)¯
NA ∞
μ5=0(μ5,a2/2)σi+μ5
λ3=0σi+μ5
λ3¯
σi+μ5−λ3
NA (1/¯
E,t,t+λ3+1/2,a,b)+∞
μ6=0
(μ6,b2/2)σi+μ6
λ4=0σi+μ6
λ4¯
σi+μ6−λ4
NA (1/¯
E,t,t+λ4+1/2,a,b) (9)
Fig. 2shows comparison of OP and ASER curves versus
transmit SNR for different number of relays (K=1 and 2) at
several threshold SNRs (γth =10 dB and 20 dB), and at differ-
ent constellation orders (M=4×2 and 32×16), respectively,
in the considered systems. We observe that the theoretical
and the simulated curves for both OP and ASER are very
close to each other. The asymptotic outage curves also align
with the theoretical and the simulated curves of OP in high
SNR regime, which further validate the analysis. Further, it is
observed that the OP performance improves by increasing the
number of relays when γth =10 dB, and remains unaffected
for γth =20 dB. This result can be validated from (5) and (6)
in Section IV, wherein, dependency of diversity gain on num-
ber of relays have been discussed. Similarly, from the ASER
performance curves, it is observed that the amplifier shows lin-
ear characteristics for small values of constellation order and
low SNRs of the links. Thus, for small constellation order and
low SNRs, the ASER performance improves significantly by
increasing the number of relays, K. However, for high constel-
lation order and high SNRs, the ASER performance remains
unaffected with variation in K.
VII. CONCLUSION
In this letter, novel closed-form expressions of OP, asymp-
totic OP behavior, ASER expression for general order RQAM
scheme, in OFDM-based nonlinear AF multiple-relay systems
over i.n.i.d. Nakagami-mchannels with modified fading and
spreading parameters using best-relay selection scheme were
proposed. The influence of number of relays on system
performances in the presence of relay nonlinearities were also
highlighted.
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